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This kind of subtlety is largely hidden from beginning students with the reliance on the power of calculators. If the foundations of a building are askew, the entire structure is compromised. The underlying ambiguities in classical trigonometry revolving around the concept of angle are an impediment to learning mathematics, weaken its logical integrity, and introduce an unnecessary element of approximation and inaccuracy into practical applications of the subject. 4. 4 Why rational trigonometry is easier The key concepts of rational trigonometry are simpler, and mathematically more natural, than those of classical trigonometry.

5 Sides, vertices and triangles The following definitions are more precise than usual. Definition A side A1 A2 ≡ {A1 , A2 } is a set with A1 and A2 points. The line of the side A1 A2 is the line A1 A2 . The side A1 A2 is a null side precisely when A1 A2 is a null line. Note that A1 and A2 are automatically distinct by the convention that repetitions are not allowed in a set. Two sides A1 A2 and A3 A4 are parallel (respectively perpendicular) precisely when A1 A2 is parallel (respectively perpendicular) to A3 A4 .

Take c to the other side, and divide both sides by the non-zero number a b c x2 + x = − . a a Step 2. Add the square of one half the coeﬃcient of x to both sides b b2 c b2 x2 + x + 2 = 2 − . a 4a 4a a Step 3. Factor the left side and simplify the right to obtain ¶2 µ b2 − 4ac b = . x+ 2a 4a2 This has a solution in x precisely when b2 − 4ac is a square, say r2 , in which case x= −b ± r . 2a Note that the usual ‘quadratic formula’ is subsumed by this derivation. Over a general field square roots do not always exist, and if they do they are generally not unique.